CHAOS THEORY
It is a field of study in mathematics, with applications in several disciplines including, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions—a response popularly referred to as the butterfly effect.
Chaotic behavior can be observed in many natural systems, such as weather and climate. This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincare maps.
This latter idea is known as sensitive dependence on initial conditions , a circumstance discovered by Edward Lorenz (who is generally credited as the first experimenter in the area of chaos) in the early 1960s.
DEFINITION:
It is the study of non linear dynamics, in which seemingly random events are actually predictable from simple deterministic equation.
Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic systems are predictable for a while and then appear to become random. The amount of time for which the behavior of a chaotic system can be effectively predicted depends on three things: * How much uncertainty we are willing to tolerate in the forecast? * How accurately we are able to measure its current state? * Which time scale is depending on the dynamics of the system?
The two main components of chaos theory are the ideas that systems - no matter how complex they may be - rely upon an underlying order, and that very simple or small systems and events can cause very complex behaviors or events.
HISTORY:
One of the first scientists to comment on chaos was Henri Poincaré(1854–1912), a late-nineteenth century French mathematician who extensively studied topology and dynamic systems. He left writings hinting at the same unpredictability in systems that Edward Lorenz (b. 1917) would study more than half a century later. Poincaré explained, "It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible." Unfortunately, the study of dynamic systems was largely ignored long after Poincaré's death.
During the early 1960s, a few scientists from various disciplines were again taking note of "odd behavior" in complex systems such as the earth's atmosphere and the human brain. One of these scientists was Edward Lorenz, a meteorologist from the Massachusetts Institute of Technology (MIT), who was experimenting with computational models of the atmosphere. In the process of his experimentation he discovered one of chaos theory's fundamental principles—the Butterfly Effect. The Butterfly Effect is named for its assertion that a butterfly flapping its wings in Tokyo can impact weather patterns in Chicago. More scientifically, the Butterfly Effect .
Another major contributor to chaos theory is Mitchell Feigenbaum (b. 1944). A physicist at the theoretical division of the Los Alamos National Laboratory starting in 1974, Feigenbaum dedicated much of his time researching chaos and trying to build mathematical formulas that might be used to explain the phenomenon Others working on related ideas (though in different disciplines) include a Berkeley, California mathematician who formed a group to study "dynamical systems" and a population biologist pushing to study strangely-complex behavior in simple biological models. During the 1970s, these scientists and others in the United States and Europe began to see beyond what appeared to be random disorder in nature (the atmosphere, wildlife populations, etc.), finding connections in erratic behavior. As recounted by James Gleick (b.1954) in Chaos, a French mathematical physicist had just made the disputable claim that turbulence in fluids might have something to do with a bizarre, infinitely-tangled abstraction he termed a "strange attractor." Stephen Smale (b. 1930), at the University of California, Berkeley, was involved in the study of "dynamical systems." He proposed a physical law that systems can behave erratically, but the erratic behavior cannot be stable. At this point, however, main-stream science was not sure what to make of these theories, and some universities and research centers deliberately avoided association with proponents of chaos theory.
QUALITIES OF A CHAOTIC SYSTEM
A chaotic system has three simple defining features: * Chaotic systems are deterministic. That is, they have some determining equation ruling their behavior. * Chaotic systems are sensitive to initial conditions. Even a very slight change in the starting point can lead to significant different outcomes. * Chaotic systems are not random, nor disorderly. Truly random systems are not chaotic. Rather, chaos has a send of order and pattern.

CHAOS THEORY CONCEPTS
There are several key terms and concepts used in chaos theory: * Butterfly effect (also called sensitivity to initial conditions): The idea that even the slightest change in the starting point can lead to greatly different results or outcomes. * Attractor: Equilibrium within the system. It represents a state to which a system finally settles. * Strange attractor: A dynamic kind of equilibrium which represents some kind of trajectory upon which a system runs from situation to situation without ever settling down.

CHAOS THEORY AND INITIAL CONDITIONS;
Chaos theory describes complex motion and the dynamics of sensitive systems. Chaotic systems are mathematically deterministic but nearly impossible to predict. Chaos is more evident in long-term systems than in short-term systems. Behavior in chaotic systems is a periodic, meaning that no variable describing the state of the system undergoes a regular repetition of values. A chaotic system can actually evolve in a way that appears to be smooth and ordered, however. Chaos refers to the issue of whether or not it is possible to make accurate long-term predictions of any system if the initial conditions are known to an accurate degree.

Chaotic systems, in this case a fractal, can appear to be smooth and ordered
Chaos occurs when a system is very sensitive to initial conditions. Initial conditions are the values of measurements at a given starting time. The phenomenon of chaotic motion was considered a mathematical oddity at the time of its discovery, but now physicists know that it is very widespread and may even be the norm in the universe. The weather is an example of a chaotic system. In order to make long-term weather forecasts it would be necessary to take an infinite number of measurements, which would be impossible to do. Also, because the atmosphere is chaotic, tiny uncertainties would eventually overwhelm any calculations and defeat the accuracy of the forecast. The presence of chaotic systems in nature seems to place a limit on our ability to apply deterministic physical laws to predict motions with any degree of certainty.
WHAT IS FRACTAL?
A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale. If the replication is exactly the same at every scale, it is called a self-similar pattern. Fractals can also be nearly the same at different levels.
A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc.

EXAMPLES OF CHAOS THEORY;

1) CHAOS THEORY IN BUSINESS:
During the 1980s, chaos theory did begin to change decision-making processes in business. A good example is the evolution of high-functioning teams. Members of effective teams frequently recreate the role each member plays, depending on the needs of the team at a given point. Though not always the formally-designated manager, informal leaders emerge in an organization not because they have been given control, but because they have a strong sense of how to address the needs of the group and its members. The most successful leaders understand that it is not the organization or the individual who is most important, but the relationship between the two. And that relationship is in constant change.
One of the most influential business writers of the 1980s and 1990s, Tom Peters (b. 1942), wrote, Thriving on Chaos: Handbook for a Management Revolution in 1987. Peters offers a strategy to help corporations deal with the uncertainty of competitive markets through customer responsiveness, fast-paced innovation, empowering personnel, and most importantly, learning to work within an environment of change. In fact, Peters asserts that we live in "a world turned upside down," and survival depends on embracing "revolution." While not explicitly concerned with chaos theory, Peters's focus on letting an organization (and its people) drive itself is quite compatible with the central tenets of chaos theory. 2) BUTTERFLY EFFECT;
Weather prediction is an extremely difficult problem. Meteorologists can predict the weather for short periods of time, a couple days at most, but beyond that predictions are generally poor.
Edward Lorenz was a mathematician and meteorologist at the Massachusetts Institute of Technology who loved the study of weather. With the advent of computers, Lorenz saw the chance to combine mathematics and meteorology. He set out to construct a mathematical model of the weather, namely a set of differential equations that represented changes in temperature, pressure, wind velocity, etc. In the end, Lorenz stripped the weather down to a crude model containing a set of 12 differential equations.
On a particular day in the winter of 1961, Lorenz wanted to re-examine a sequence of data coming from his model. Instead of restarting the entire run, he decided to save time and restart the run from somewhere in the middle. Using data printouts, he entered the conditions at some point near the middle of the previous run, and re-started the model calculation. What he found was very unusual and unexpected. The data from the second run should have exactly matched the data from the first run. While they matched at first, the runs eventually began to diverge dramatically — the second run losing all resemblance to the first within a few "model" months. A sample of the data from his two runs in shown below:

At first Lorenz thought that a vacuum tube had gone bad in his computer, a Royal McBee — an extremely slow and crude machine by today's standards. After discovering that there was no malfunction, Lorenz finally found the source of the problem. To save space, his printouts only showed three digits while the data in the computer's memory contained six digits. Lorenz had entered the rounded-off data from the printouts assuming that the difference was inconsequential. For example, even today temperature is not routinely measured within one part in a thousand.
This led Lorenz to realize that long-term weather forecasting was doomed. His simple model exhibits the phenomenon known as "sensitive dependence on initial conditions." This is sometimes referred to as the butterfly effect, e.g. a butterfly flapping its wings in South America can affect the weather in Central Park. The question then arises why does a set of completely deterministic equations exhibit this behavior? After all, scientists are often taught that small initial perturbations lead to small changes in behavior. This was clearly not the case in Lorenz's model of the weather. The answer lies in the nature of the equations; they were nonlinear equations. While they are difficult to solve, nonlinear systems are central to chaos theory and often exhibit fantastically complex and chaotic behavior.
Applications;
Chaos theory was born from observing weather patterns, but it has become applicable to a variety of other situations. Some areas benefiting from chaos theory today are geology, mathematics, microbiology, biology, computer science,economics,,finance, algorithmic trading, meteorology, philosophy, physics, politics, population dynamics, psychology, and robotics. A few categories are listed below with examples, but this is by no means a comprehensive list as new applications are appearing every day.
Computer science
Chaos theory is not new to computer science and has been used for many years in cryptography. One type of encryption, secret key or symmetric key, relies on diffusion and confusion, which is modeled well by chaos theory.Another type of computing, DNA computing, when paired with chaos theory, offers a more efficient way to encrypt images and other information. Robotics is another area that has recently benefited from chaos theory. Instead of robots acting in a trial-and-error type of refinement to interact with their environment, chaos theory has been used to build a predictive model.
Biology
For over a hundred years, biologists have been keeping track of populations of different species with population models. Most models are deterministic systems, but recently scientists have been able to implement chaotic models in certain populations. For example, a study on models of Canadian lynx showed there was chaotic behavior in the population growth. Chaos can also be found in ecological systems, such as hydrology. While a chaotic model for hydrology has its shortcomings, there is still much to be learned from looking at the data through the lens of chaos theory. Another biological application is found in cardiotocography. Fetal surveillance is a delicate balance of obtaining accurate information while being as noninvasive as possible. Better models of warning signs of fetal hypoxia can be obtained through chaotic modeling

Social sciences: In the social sciences, chaos theory is the study of complex non-linear systems of social complexity. It is not about disorder, but rather is about very complicated systems of order.
Nature, including some instances of social behavior and social systems, is highly complex, and the only prediction you can make is that it is unpredictable. Chaos theory looks at this unpredictability of nature and tries to make sense of it.
Chaos theory aims to find the general order of social systems, and particularly social systems that are similar to each other. The assumption here is that the unpredictability in a system can be represented as overall behavior, which gives some amount of predictability, even when the system is unstable. Chaotic systems are not random systems. Chaotic systems have some kind of order, with an equation that determines overall behavior.
Chaos in psychology:

Chaos theory has successfully explained various phenomena in the natural sciences and has subsequently been heralded by some as the new paradigm for science. Chaos and its concepts are beginning to be applied to psychology by researchers from cognitive, developmental and clinical psychology. This paper seeks to provide an overview of this work and evaluate the application of chaos to psychology. Chaos is briefly explained before existing applications of chaos in psychology and possible implications are examined. Finally, problems of applying chaos are evaluated and conclusions drawn regarding the usefulness of chaos in psychology.

Other areas:
The principles of Chaos Theory have been successfully used to describe and explain diverse natural and artificial phenomena. Such as:

* Predicting epileptic seizures.
* Predicting financial markets.
* Modeling of manufacturing systems.
* Making weather forecasts.
* Creating Fractals. Computer-generated images applying Chaos Theory principles.

In a scenario where businesses operate in a turbulent, complex and unpredictable environment, the tenets of Chaos Theory can be extremely valuable. Application areas can include:

* Business Strategy / Corporate Strategy.
* Complex decision-making
* Organizational behavior and organizational change. Compare: Causal Model of Organizational Performance and Change
* Stock market behavior, investing.
STRANGE ATTRACTORS In dynamical systems, an attractor is a set of physical properties toward which a system tends to evolve, regardless of the starting conditions of the system.[1] Property values that get close enough to the attractor values remain close even if slightly disturbed.In finite-dimensional systems, the evolving variable may be represented algebraically as an n-dimensional vector. The attractor is a region in n-dimensional space. In physical systems, the n dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in economic systems, they may be separate variables such as the inflation rateand the unemployment rate.If the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented geometrically in two or three dimensions, (as for example in the three-dimensional case depicted to the right). An attractor can be a point, a finite set of points, a curve, a manifold, or even a complicated set with afractal structure known as a strange attractor. If the variable is a scalar, the attractor is a subset of the real number line. Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory.History:Edward Lorenz's first weather model exhibited chaotic behavior, but it involved a set of 12 nonlinear differential equations. Lorenz decided to look for complex behavior in an even simpler set of equations, Lorenz simplified a few fluid dynamics equations (called the Navier-Stokes equations) and ended up with a set of three nonlinear equations:where P is the Prandtl number representing the ratio of the fluid viscosity to its thermal conductivity, R represents the difference in temperature between the top and bottom of the system, and B is the ratio of the width to height of the box used to hold the system. The values Lorenz used are P = 10, R = 28, B = 8/3.On the surface these three equations seem simple to solve. However, they represent an extremely complicated dynamical system. If one plots the results in three dimensions the following figure, called the Lorenz attractor, is obtained.Projections of this attractor in the y-z and x-z two-dimensional planes are as follows:Projection on the y-z planeProjection on the x-z planeThe Lorenz attractor is an example of a strange attractor. Strange attractors are unique from other phase-space attractors in that one does not know exactly where on the attractor the system will be. Two points on the attractor that are near each other at one time will be arbitrarily far apart at later times. The only restriction is that the state of system remain on the attractor. Strange attractors are also unique in that they never close on themselves — the motion of the system never repeats (non-periodic). The motion we are describing on these strange attractors is what we mean by chaotic behavior.The Lorenz attractor was the first strange attractor, but there are many systems of equations that give rise to chaotic dynamics. Examples of other strange attractors include the Rössler and Hénon attractors.Definition:A strange attractor is a concept in chaos theory that is used to describe the behavior of chaotic systems. Unlike a normal attractor, a strange attractor predicts the formation of semi-stable patterns that lack a fixed spatial position. An equation that includes a strange attractor must incorporate non-integer dimensional values, resulting in a pattern of trajectories that seem to appear randomly within the system. Strange attractors appear in both natural and theoretical diagrams of phase space models.An attractor is a component in a dynamic system that increases the likelihood that other components will draw closer to a specific field or point when they approach within a certain distance of the attractor. After they have passed within a certain distance of the attractor, these components will adopt a stable configuration and resist minor disturbances in the system. For example, the lowest point in the arc of a pendulum is a simple attractor. A phase space model of a pendulum will chart a series of points growing closer to the low point each time their trajectory takes them past it, until they cluster around the low | |
Example of strange attarctors;
A second example of a strange attractor is customization/differentiation (or diversity). On the district level, each school has the freedom to be different from other schools. On the building level each teacher has the freedom to be different from other teachers. And on the classroom level each student has the freedom to be different from other students (with respect to both what to learn and how to learn it). A third example is shared decisionmaking/collaboration.
On the district level the school board and superintendent involve community members, teachers, and staff in policymaking and decision making. On the school level the principal involves parents, teachers,and staff in policymaking and decision making. And on the classroom level the teacher involves the child and parents in decisions and activities to promote the child’s learning and development.
To become an effective strange attractor for the transformation of a school system , the core ideas and values(or beliefs) must become fairly widespread cultural norms among the stakeholders most involved with making the changes.
Once that status is reached, verylittle planning needs to be done for the transformation to take place. Appropriate behaviors and structures will emerge spontaneously through a process called self-organization.

One example of a fractal in education is autocratic control. On the school board typically controls the superintendent. On the district level, the superintendent controls the principals.
On the building level the principals control their teachers. And on the classroom level the teachers control their students.

Applications to Appreciative Inquiry

Appreciative Inquiry techniques, as outlined in Watkins' book, implement the third of these approaches to shift attractors. The five generic processes operationalize the idea of shifting an attractor by using points within the existing attractor as seeds for a new pattern.
Choose the positive as focus of inquiry. This process examines the existing attractor patterns to recognize points that can serve as seeds for the new.
Inquire into life-giving forces. This articulates possible attractor regimes that can be encouraged in future. It begins to tie together the positive points into coherent wholes in ways that people can see and understand them.
Locate themes that appear in the stories and select topics for further inquiry. This pulls the emergent patterns out of the stories. These clear and concise patterns become the foundation for the emerging attractor.
Create shared images for a preferred future. This articulates the future attractor, which includes the points and patterns that emerged from earlier steps. This becomes, then the new story that will constrain and shape the agents' behaviors as they move into the new attractor regime.
Find innovative ways to create that future. This is the process of identifying reinforcing loops to inform and constrain individual and group behavior into the new, intended attractor.

Applications of Strange Attractors to Human System Behavior ?
Quantitative research has articulated the strange attractors that shape a variety of dynamical human systems, including conversations (Eoyang and Stewart), accident rates (Guastello), economics (Kiel). Such quantitative analysis requires that the systems incorporate a small number of deterministic variables (dimensions). If the dimensionality of the system is too high (the commonly-used limit is eight variables), the system is considered to be random because the pattern cannot be discerned by current manipulative practices and analytical algorithms.
Qualitatively, however, the strange attractor has been used as a metaphor to describe highly complex, but patterned, behavior in human systems. Whenever the behavior of the system is bounded, includes infinite freedom within the bounds, and generates coherent patterns over time, the human system can be metaphorically described as a strange attractor regime. Examples of human system aspects that fit this qualitative description include organizational culture, patterns of professional practice, or the behaviors of firms within a given industry. In each case, individual agents work within accepted boundaries in accord with patterns of behavior that are supported by the rest of the system in complex and nonlinear ways.
Working in the metaphorical realm, it is seductive to think of the attractor as a reified object, which shapes the behavior of the system agents. Any agent in the system is forced by multiple system influences into individual patterns of behavior that match those of other agents and the system as a whole. This approach can be helpful when the purpose of the description is to articulate the current system patterns and to constrain the behavior of individuals to conform to the existing patterns, however complex.
Such a view, however, can be counter-productive when the purpose of the description is to change existing patterns on an organizational or individual level of action. If the desire is to change the attractor, then one must deconstruct the current and investigate ways to establish new attractors to shape new patterns of behavior.

* http://www.referenceforbusiness.com/management/Bun-Comp/Chaos-Theory.html#ixzz3F5qVh2b3 * http://fractalfoundation.org/resources/what-are-fractals/ * http://www.indiana.edu/~syschang/decatur/documents/chaos_reigeluth_s2004.pdf * http://www.tnellen.com/alt/chaos.html * http://www.stsci.edu/~lbradley/seminar/butterfly.html * http://tap.sagepub.com/content/7/3/373.short * http://allaboutchaostheory.blogspot.com/2009/02/applications-of-chaos-theory.html * http://www.scienceclarified.com/Ca-Ch/Chaos-Theory.html#ixzz3Ensl75d2 * http://www.stsci.edu/~lbradley/seminar/attractors.html * http://www.referenceforbusiness.com/management/Bun-Comp/Chaos-Theory.html * http://chaostheoryinnursing.blogspot.com/ * http://www.wisegeek.com/what-is-a-strange-attractor.htm * *